Positivity of the second exterior power of the tangent bundles

TL;DR

This paper investigates the structure of smooth complex projective varieties with nef second exterior power of the tangent bundle, revealing their fibrations, contractions, and conditions under which they are Fano or blow-ups of projective space.

Contribution

It establishes a structure theorem for such varieties, showing they are fibered over abelian varieties with Fano fibers and classifies certain contractions, including blow-ups of projective space.

Findings

Up to finite étale cover, the Albanese map is a locally trivial fibration with Fano fibers.

Either the tangent bundle is nef or the variety is Fano.

Contractions of $K_X$-negative extremal rays are classified, including blow-ups of projective space.

Abstract

Let $X$ be a smooth complex projective variety with nef $\bigwedge^2 T_X$ and $\dim X \geq 3$. We prove that, up to a finite \'etale cover $\tilde{X} \to X$, the Albanese map $\tilde{X} \to {\rm Alb}(\tilde{X})$ is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety $F$ with nef $\bigwedge^2 T_F$. As a bi-product, we see that either $T_X$ is nef or $X$ is a Fano variety. Moreover we study a contraction of a $K_X$-negative extremal ray $\varphi: X \to Y$. In particular, we prove that $X$ is isomorphic to the blow-up of a projective space at a point if $\varphi$ is of birational type. We also prove that $\varphi$ is a smooth morphism if $\varphi$ is of fiber type. As a consequence, we give a structure theorem of varieties with nef $\bigwedge^2 T_X$.